getdoc3129. 452KB Jun 04 2011 12:04:13 AM
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Vol. 13 (2008), Paper no. 22, pages 629–670.
Journal URL
http://www.math.washington.edu/~ejpecp/
The non-linear stochastic wave equation in high
dimensions
Daniel Conus and Robert C. Dalang∗
Institut de math´ematiques
Ecole Polytechnique F´ed´erale de Lausanne
Station 8
CH-1015 Lausanne
Switzerland
daniel.conus@epfl.ch, robert.dalang@epfl.ch
Abstract
We propose an extension of Walsh’s classical martingale measure stochastic integral that
makes it possible to integrate a general class of Schwartz distributions, which contains the
fundamental solution of the wave equation, even in dimensions greater than 3. This leads to
a square-integrable random-field solution to the non-linear stochastic wave equation in any
dimension, in the case of a driving noise that is white in time and correlated in space. In the
particular case of an affine multiplicative noise, we obtain estimates on p-th moments of the
solution (p > 1), and we show that the solution is H¨older continuous. The H¨older exponent
that we obtain is optimal .
Key words: Martingale measures, stochastic integration, stochastic wave equation, stochastic partial differential equations, moment formulae, H¨older continuity.
AMS 2000 Subject Classification: Primary 60H15; Secondary: 60H20, 60H05.
Submitted to EJP on February 21, 2007, final version accepted April 1, 2008.
∗
Partially supported by the Swiss National Foundation for Scientific Research.
629
1
Introduction
In this paper, we are interested in random field solutions to the stochastic wave equation
∂2
u(t, x) − ∆u(t, x) = α(u(t, x))F˙ (t, x) + β(u(t, x)),
∂t2
t > 0, x ∈ Rd ,
(1.1)
with vanishing initial conditions. In this equation, d > 1, ∆ denotes the Laplacian on Rd , the
functions α, β : R → R are Lipschitz continuous and F˙ is a spatially homogeneous Gaussian
noise that is white in time. Informally, the covariance functional of F˙ is given by
E[F˙ (t, x)F˙ (s, y)] = δ(t − s)f (x − y),
s, t > 0, x, y ∈ Rd ,
where δ denotes the Dirac delta function and f : Rd → R+ is continuous on Rd \ {0} and even.
We recall that a random field solution to (1.1) is a family of random variables (u(t, x), t ∈
R+ , x ∈ Rd ) such that (t, x) 7→ u(t, x) from R+ × Rd into L2 (Ω) is continuous and solves
an integral form of (1.1): see Section 4. Having a random field solution is interesting if, for
instance, one wants to study the probability density function of the random variable u(t, x)
for each (t, x), as in [12]. A different notion is the notion of function-valued solution, which
is a process t → u(t) with values in a space such as L2 (Ω, L2loc (Rd , dx)) (see for instance [7],
[4]). In some cases, such as [6], a random field solution can be obtained from a function-valued
solution by establishing (H¨
older) continuity properties of (t, x) 7→ u(t, x), but such results are
not available for the stochastic wave equation in dimensions d > 4. In other cases (see [3]), the
two notions are genuinely distinct (since the latter would correspond to (t, x) 7→ u(t, x) from
R+ × Rd into L2 (Ω) is merely measurable), and one type of solution may exist but not the other.
We recall that function-valued solutions to (1.1) have been obtained in all dimensions [14] and
that random field solutions have only been shown to exist when d ∈ {1, 2, 3} (see [1]).
In spatial dimension 1, a solution to the non-linear wave equation driven by space-time white
noise was given in [24], using Walsh’s martingale measure stochastic integral. In dimensions 2
or higher, there is no function-valued solution with space-time white noise as a random input:
some spatial correlation is needed in this case. In spatial dimension 2, a necessary and sufficient
condition on the spatial correlation for existence of a random field solution was given in [2].
Study of the probability law of the solution is carried out in [12].
In spatial dimension d = 3, existence of a random field solution to (1.1) is given in [1]. Since the
fundamental solution in this dimension is not a function, this required an extension of Walsh’s
martingale measure stochastic integral to integrands that are (Schwartz) distributions. This
extension has nice properties when the integrand is a non-negative measure, as is the case for
the fundamental solution of the wave equation when d = 3. The solution constructed in [1] had
moments of all orders but no spatial sample path regularity was established. Absolute continuity
and smoothness of the probability law was studied in [16] and [17] (see also the recent paper
[13]). H¨older continuity of the solution was only recently established in [6], and sharp exponents
were also obtained.
In spatial dimension d > 4, random field solutions were only known to exist in the case of the
linear wave equation (α ≡ 1, β ≡ 0). The methods used in dimension 3 do not apply to higher
dimensions, because for d > 4, the fundamental solution of the wave equation is not a measure,
but a Schwartz distribution that is a derivative of some order of a measure (see Section 5). It
630
was therefore not even clear that the solution to (1.1) should be H¨older continuous, even though
this is known to be the case for the linear equation (see [20]), under natural assumptions on the
covariance function f .
In this paper, we first extend (in Section 3) the construction of the stochastic integral given in
[1], so as to be able to define
Z tZ
S(s, x)Z(s, x)M (ds, dx)
Rd
0
in the case where M (ds, dx) is the martingale measure associated with the Gaussian noise F˙ ,
Z(s, x) is an L2 -valued random field with spatially homogeneous covariance, and S is a Schwartz
distribution, that is not necessarily non-negative (as it was in [1]). Among other technical
conditions, S must satisfy the following condition, that also appears in [14]:
Z
Z t
ds sup
µ(dξ) |FS(s)(ξ + η)|2 < ∞,
0
η∈Rd
Rd
where µ is the spectral measure of F˙ (that is, Fµ = f , where F denotes the Fourier transform).
With this stochastic integral, we can establish (in Section 4) existence of a random field solution
of a wide class of stochastic partial differential equations (s.p.d.e.’s), that contains (1.1) as a
special case, in all spatial dimensions d (see Section 5).
However, for d > 4, we do not know in general if this solution has moments of all orders.
We recall that higher order moments, and, in particular, estimates on high order moments of
increments of a process, are needed for instance to apply Kolmogorov’s continuity theorem and
obtain H¨older continuity of sample paths of the solution.
In Section 6, we consider the special case where α is an affine function and β ≡ 0. This is
analogous to the hyperbolic Anderson problem considered in [5] for d 6 3. In this case, we
show that the solution to (1.1) has moments of all orders, by using a series representation of the
solution in terms of iterated stochastic integrals of the type defined in Section 3.
Finally, in Section 7, we use the results of Section 6 to establish H¨older continuity of the solution
to (1.1) (Propositions 7.1 and 7.2) for α affine and β ≡ 0. In the case where the covariance
function is a Riesz kernel, we obtain the optimal H¨older exponent, which turns out to be the
same as that obtained in [6] for dimension 3.
2
Framework
In this section, we recall the framework in which the stochastic integral is defined. We consider
a Gaussian noise F˙ , white in time and correlated in space. Its covariance function is informally
given by
E[F˙ (t, x)F˙ (s, y)] = δ(t − s)f (x − y),
s, t > 0, x, y ∈ Rd ,
where δ stands for the Dirac delta function and f : Rd → R+ is continuous on Rd \ {0} and
even. Formally, let D(Rd+1 ) be the space of C ∞ -functions with compact support and let F =
{F (ϕ), ϕ ∈ D(Rd+1 )} be an L2 (Ω, F, P)-valued mean zero Gaussian process with covariance
functional
Z
Z ∞ Z
dy ϕ(t, x)f (x − y)ψ(t, y).
dx
dt
E[F (ϕ)F (ψ)] =
0
Rd
Rd
631
Since f is a covariance, there exists a non-negative tempered measure µ whose Fourier transform
is f . That is, for all φ ∈ S(Rd ), the Schwartz space of C ∞ -functions with rapid decrease, we
have
Z
Z
Fφ(ξ)µ(dξ).
f (x)φ(x)dx =
Rd
Rd
As f is the Fourier transform of a tempered measure, it satisfies an integrability condition of
the form
Z
f (x)
dx < ∞,
(2.1)
1
+ |x|p
d
R
for some p < ∞ (see [21, Theorem XIII, p.251]).
Following [2], we extend this process to a worthy martingale measure M = (Mt (B), t > 0, B ∈
Bb (Rd )), where Bb (Rd ) denotes the bounded Borel subsets of R, in such a way that for all
ϕ ∈ S(Rd+1 ),
Z ∞Z
ϕ(t, x)M (dt, dx),
F (ϕ) =
Rd
0
where the stochastic integral is Walsh’s stochastic integral with respect to the martingale measure
M (see [24]). The covariation and dominating measure Q and K of M are given by
Q([0, t] × A × B) = K([0, t] × A × B)
= hM (A), M (B)it = t
Z
Rd
dx
Z
dy 1A (x)f (x − y)1B (y).
Rd
We consider the filtration Ft given by Ft = Ft0 ∨ N , where
Ft0 = σ(Ms (B), s 6 t, B ∈ Bb (Rd ))
and N is the σ-field generated by the P-null sets.
Fix T > 0. The stochastic integral of predictable functions g : R+ × Rd × Ω → R such that
kgk+ < ∞, where
kgk2+ = E
·Z
T
ds
Z
dx
Rd
0
Z
Rd
¸
dy |g(s, x, ·)| f (x − y) |g(s, y, ·)| ,
is defined by Walsh (see [24]). The set of such functions is denoted by P+ . Dalang [1] then
introduced the norm k · k0 defined by
kgk20 = E
·Z
T
ds
0
Z
Rd
dx
Z
Rd
¸
dy g(s, x, ·)f (x − y)g(s, y, ·) .
(2.2)
Recall that a function g is called elementary if it is of the form
g(s, x, ω) = 1]a,b] (s)1A (x)X(ω),
(2.3)
where 0 6 a < b 6 T , A ∈ Bb (Rd ), and X is a bounded Fa -measurable random variable. Now
let E be the set of simple functions, i.e., the set of all finite linear combinations of elementary
functions. Since the set of predictable functions such that kgk0 < ∞ is not complete, let P0
632
denote the completion of the set of simple predictable functions with respect to k · k0 . Clearly,
P+ ⊂ P0 . Both P0 and P+ can be identified with subspaces of P, where
n
P :=
t 7→ S(t) from [0, T ] × Ω → S ′ (Rd ) predictable, such that FS(t) is a.s.
a function and kSk0 < ∞} ,
where
kSk20
=E
·Z
T
dt
Z
2
µ(dξ) |FS(t)(ξ)|
Rd
0
¸
(2.4)
.
For S(t) ∈ S(Rd ), elementary properties of convolution and Fourier transform show that (2.2)
and (2.4) are equal. When d > 4, the fundamental solution of the wave equation provides an
example of an element of P0 that is not in P+ (see Section 5).
Consider a predictable process (Z(t, x), 0 6 t 6 T, x ∈ Rd ), such that
sup sup E[Z(t, x)2 ] < ∞.
06t6T x∈Rd
Let M Z be the martingale measure defined by
Z tZ
Z(s, y)M (ds, dy),
MtZ (B) =
0
0 6 t 6 T, B ∈ Bb (Rd ),
B
in which we again use Walsh’s stochastic integral [24]. We would like to give a meaning to
the stochastic integral of a large class of S ∈ P with respect to the martingale measure M Z .
Following the same idea as before, we will consider the norms k · k+,Z and k · k0,Z defined by
¸
·Z T Z
Z
2
dy |g(s, x, ·)Z(s, x)f (x − y)Z(s, y)g(s, y, ·)|
dx
kgk+,Z = E
ds
and
kgk20,Z
=E
·Z
T
ds
0
Rd
Rd
0
Z
dx
Rd
Z
¸
dy g(s, x, ·)Z(s, x)f (x − y)Z(s, y)g(s, y, ·) .
Rd
(2.5)
Let P+,Z be the set of predictable functions g such that kgk+,Z < ∞. The space P0,Z is
defined, similarly to P0 , as the completion of the set of simple predictable functions, but taking
completion with respect to k · k0,Z instead of k · k0 .
For g ∈ E, as in (2.3), the stochastic integral g · M Z = ((g · M Z )t , 0 6 t 6 T ) is the squareintegrable martingale
Z tZ
Z
Z
g(s, y, ·)Z(s, y)M (ds, dy).
(g · M Z )t = Mt∧b
(A) − Mt∧a
(A) =
0
Rd
Notice that the map g 7→ g ·M Z , from (E, k·k0,Z ) into the Hilbert space M of continuous square1
integrable (Ft )-martingales X = (Xt , 0 6 t 6 T ) equipped with the norm kXk = E[XT2 ] 2 , is an
isometry. Therefore, this isometry can be extended to an isometry S 7→ S·M Z from (P0,Z , k·k0,Z )
into M. The square-integrable martingale S · M Z = ((S · M Z )t , 0 6 t 6 T ) is the stochastic
integral process of S with respect to M Z . We use the notation
Z tZ
S(s, y)Z(s, y)M (ds, dy)
0
Rd
for (S · M Z )t .
The main issue is to identify elements of P0,Z . We address this question in the next section.
633
3
Stochastic Integration
In this section, we extend Dalang’s result concerning the class of Schwartz distributions for which
the stochastic integral with respect to the martingale measure M Z can be defined, by deriving
a new inequality for this integral. In particular, contrary to [1, Theorem 2], the result presented
here does not require that the Schwartz distribution be non-negative.
In Theorem 3.1 below, we show that the non-negativity assumption can be removed provided
the spectral measure satisfies the condition (3.6) below, which already appears in [14] and [4].
As in [1, Theorem 3], an additional assumption similar to [1, (33), p.12] is needed (hypothesis
(H2) below). This hypothesis can be replaced by an integrability condition (hypothesis (H1)
below).
Suppose Z is a process such that sup06s6T E[Z(s, 0)2 ] < +∞ and with spatially homogeneous
covariance, that is z 7→ E[Z(t, x)Z(t, x + z)] does not depend on x. Following [1, Theorem 3],
set f Z (s, x) = f (x)gs (x), where gs (x) = E[Z(s, 0)Z(s, x)].
For s fixed, the function gs is non-negative definite, since it is a covariance function. Hence,
there exists a non-negative tempered measure νsZ such that gs = FνsZ . Note that νsZ (Rd ) =
gs (0) = E[Z(s, 0)2 ]. Using the convolution property of the Fourier transform, we have
f Z (s, ·) = f · gs = Fµ · FνsZ = F(µ ∗ νsZ ),
where ∗ denotes convolution. Looking back to the definition of k · k0,Z , we obtain, for a deterministic ϕ ∈ P0,Z with ϕ(t, ·) ∈ S(Rd ) for all 0 6 t 6 T (see [1, p.10]),
Z T Z
Z
kϕk20,Z =
ds
dx
dy ϕ(s, x)f (x − y)gs (x − y)ϕ(s, y)
0
Rd
Rd
Z T Z
(µ ∗ νsZ )(dξ) |Fϕ(s, ·)(ξ)|2
ds
=
d
R
0
Z
Z T Z
Z
µ(dξ) |Fϕ(s, ·)(ξ + η)|2 .
(3.1)
νs (dη)
ds
=
Rd
Rd
0
In particular,
kϕk20,Z
6
Z
T
0
6 C
Z
ds νsZ (Rd )
T
ds sup
0
η∈Rd
sup
η∈Rd
Z
Z
µ(dξ) |Fϕ(s, ·)(ξ + η)|2
Rd
µ(dξ) |Fϕ(s, ·)(ξ + η)|2 ,
(3.2)
Rd
where C = sup06s6T E[Z(s, 0)2 ] < ∞ by assumption. Taking (3.1) as the definition of k · k0,Z ,
we can extend this norm to the set P Z , where
n
P Z :=
t 7→ S(t) from [0, T ] → S ′ (Rd ) deterministic, such that FS(t) is
a function and kSk0,Z < ∞} .
The spaces P+,Z and P0,Z will now be considered as subspaces of P Z . Let S ∈ P Z . We will
need the following two hypotheses to state the next theorem. Let B(0, 1) denote the open ball
in Rd that is centered at 0 with radius 1.
634
(H1) For all ϕ ∈ D(Rd ) such that ϕ > 0, supp(ϕ) ⊂ B(0, 1), and
0 6 a 6 b 6 T , we have
Z b
(S(t) ∗ ϕ)(·) dt ∈ S(Rd ),
R
Rd
ϕ(x)dx = 1, and for all
(3.3)
a
and
Z
dx
Rd
Z
T
ds |(S(s) ∗ ϕ)(x)| < ∞.
(3.4)
0
(H2) The function FS(t) is such that
lim
h↓0
Z
T
ds sup
η∈Rd
0
Z
|FS(r)(ξ + η) − FS(s)(ξ + η)|2 = 0.
sup
µ(dξ)
Rd
(3.5)
s 1, take
ψn (x) = nd ψ(nx). Then ψn → δ0 in S ′ (Rd ) as n → ∞. Moreover, Fψn (ξ) = Fψ( nξ ) and
|Fψn (ξ)| 6 1, for all ξ ∈ Rd . Define Sn (t) = (ψn ∗ S)(t). As S(t) is of rapid decrease, we have
Sn (t) ∈ S(Rd ) (see [21], Chap. VII, §5, p.245).
Suppose that Sn ∈ P0,Z for all n. Then
kSn −
Sk20,Z
=
Z
T
ds
Rd
0
=
Z
0
Z
T
ds
Z
Rd
νsZ (dη)
νsZ (dη)
Z
Z
µ(dξ) |F(Sn (s) − S(s))(ξ + η)|2
Rd
µ(dξ) |Fψn (ξ + η) − 1|2 |FS(s)(ξ + η)|2 .
Rd
635
(3.8)
The expression |Fψn (ξ + η) − 1|2 is bounded by 4 and goes to 0 as n → ∞ for every ξ and η. By
(3.6), the Dominated Convergence Theorem shows that kSn − Sk0,Z → 0 as n → ∞. As P0,Z is
complete, if Sn ∈ P0,Z for all n, then S ∈ P0,Z .
To complete the proof, it remains to show that Sn ∈ P0,Z for all n.
First consider assumption (H2). In this case, the proof that Sn ∈ P0,Z is based on the same
approximation as in [1]. For n fixed, we can write Sn (t, x) because Sn (t) ∈ S(Rd ) for all
0 6 t 6 T . The idea is to approximate Sn by a sequence of elements of P+,Z . For all m > 1, set
Sn,m (t, x) =
m −1
2X
Sn (tk+1
(t),
m , x)1[tk ,tk+1
m [
m
(3.9)
k=0
where tkm = kT 2−m . Then Sn,m (t, ·) ∈ S(Rd ). We now show that Sn,m ∈ P+,Z . Being a
deterministic function, Sn,m is predictable. Moreover, using the definition of k · k+,Z and the
fact that |gs (x)| 6 C for all s and x, we have
Z
Z T Z
2
dy |Sn,m (s, x)| f (x − y) |gs (x − y)||Sn,m (s, y)|
dx
ds
kSn,m k+,Z =
=
Rd
0
m −1 Z k+1
2X
tm
Rd
ds
tkm
k=0
m −1 Z k+1
2X
tm
6 C
k=0
tkm
Z
dx
Rd
Rd
ds
Z
Z
Rd
k+1
dy |Sn (tk+1
m , x)| f (x − y) |gs (x − y)| |Sn (tm , y)|
˜ k+1
dz f (z)(|Sn (tk+1
m , ·)| ∗ |Sn (tm , ·)|)(z),
k+1
where S˜n (tk+1
m , x) = Sn (tm , −x). By Leibnitz’ formula (see [22], Ex. 26.4, p.283), the function
k+1
k+1
˜
z 7→ (|Sn (tm , ·)| ∗ |Sn (tm , ·)|)(z) decreases faster than any polynomial in |z|−1 . Therefore, by
(2.1), the preceding expression is finite and kSn,m k+,Z < ∞, and Sn,m ∈ P+,Z ⊂ P0,Z .
The sequence of elements of P+,Z that we have constructed converges in k · k0,Z to Sn . Indeed,
Z
Z T Z
µ(dξ) |F(Sn,m (s, ·) − Sn (s, ·))(ξ + η)|2
νsZ (dη)
ds
kSn,m − Sn k20,Z =
d
d
R
R
0
Z
Z T Z
µ(dξ)
sup
νsZ (dη)
ds
|F(Sn (r, ·) − Sn (s, ·))(ξ + η)|2 ,
6
0
Rd
Rd
s0 is a martingale. Moreover,
2
sup Eµ(dξ)×νsZ (dη)×ds [Xm
] 6 Eµ(dξ)×νsZ (dη)×ds [X 2 ] < ∞.
m
The martingale L2 -convergence theorem then shows that (3.12) goes to 0 as m → ∞ and hence
that Sn ∈ P0,Z .
Now, by the isometry property of the stochastic integral between P0,Z and the set M2 of squareintegrable martingales, (S · M Z )t is well-defined and
E[(S · M Z )2T ] = kSk20,Z =
Z
T
ds
0
Z
Rd
νsZ (dη)
Z
µ(dξ) |FS(s, ·)(ξ + η)|2 .
Rd
The bound in the second part of (3.7) is obtained as in (3.2). The result is proved.
¥
Remark 3.2. As can be seen by inspecting the proof, Theorem 3.1 is still valid if we replace
(H2) by the following assumptions :
• t 7→ FS(t)(ξ) is continuous in t for all ξ ∈ Rd ;
• there exists a function t 7→ k(t) with values in the space Sr′ (Rd ) such that, for all 0 6 t 6 T
and h ∈ [0, ε],
|FS(t + h)(ξ) − FS(t)(ξ)| 6 |Fk(t)(ξ)|,
and
Z
T
ds sup
0
η∈Rd
Z
µ(dξ) |Fk(s)(ξ + η)|2 < +∞.
Rd
Remark 3.3. There are two limitations to our construction of the stochastic integral in Theorem
3.1. The first concerns stationarity of the covariance of Z. Under certain conditions (which, in
the case where S is the fundamental solution of the wave equation, only hold for d 6 3), Nualart
and Quer-Sardanyons [13] have removed this assumption. The second concerns positivity of the
covariance function f . A weaker condition appears in [14], where function-valued solutions are
studied.
Integration with respect to Lebesgue measure
In addition to the stochastic integral defined above, we will have to define the integral of the
product of a Schwartz distribution and a spatially homogeneous process with respect to Lebesgue
measure. More precisely, we have to give a precise definition to the process informally given by
Z t Z
dy S(s, y)Z(s, y),
ds
t 7→
0
Rd
where t 7→ S(t) is a deterministic function with values is the space of Schwartz distributions
with rapid decrease and Z is a stochastic process, both satisfying the assumptions of Theorem
3.1.
638
In addition, suppose first that S ∈ L2 ([0, T ], L1 (Rd )). By H¨older’s inequality, we have
"µZ
¶2 #
Z
T
dx |S(s, x)||Z(s, x)|
E
ds
0
6 CE
6 C
6 C
Z
Z
"Z
Rd
T
ds
dx |S(s, x)||Z(s, x)|
Rd
0
T
ds
0
T
ds
0
µZ
Z
Z
dx |S(s, x)|
Z
¶2 #
dy |S(s, y)| E[|Z(s, x)||Z(s, y)|]
Rd
Rd
dx |S(s, x)|
Z
dy |S(s, y)| < ∞,
(3.13)
Rd
Rd
RT R
by the assumptions on Z. Hence 0 ds Rd dx |S(s, x)||Z(s, x)| < ∞ a.s. and the process
Z t Z
dx S(s, x)Z(s, x),
t > 0,
ds
Rd
0
is a.s. well-defined as a Lebesgue-integral. Moreover,
"µZ
¶2 #
Z
T
0 6 E
ds
dx S(s, x)Z(s, x)
=
Z
T
ds
Z
T
ds
=
0
dx
Z
T
ds
Z
Rd
Z
dy S(s, x)S(s, y) E[Z(s, x)Z(s, y)]
Rd
dx
Rd
0
Z
Z
Rd
0
=
Rd
0
Z
dy S(s, x)S(s, y)gs (x − y)
Rd
νsZ (dη) |FS(s)(η)|2 ,
(3.14)
where νsZ is the measure such that FνsZ = gs . Let us define a norm k · k1,Z on the space P Z by
Z T Z
νsZ (dη) |FS(s)(η)|2 .
(3.15)
ds
kSk21,Z =
Rd
0
This norm is similar to k·k0,Z , but with µ(dξ) = δ0 (dξ). In order to establish the next proposition,
we will need the following assumption.
(H2*) The function FS(s) is such that
Z T
ds sup sup
lim
h↓0
0
|FS(r)(η) − FS(s)(η)|2 = 0.
(3.16)
η∈Rd s0 is not necessarily positive and the argument
above does not apply. We need to know a priori that the processes Z(t, x) = α(un (t, x)) −
α(v(t, x)) and W (t, x) = β(un (t, x)) − β(v(t, x)) have a spatially homogeneous covariance. This
is why we consider the restricted class of processes satisfying property “S”.
As u0 ≡ 0, it is clear that the joint process (u0 (t, x), v(t, x), t > 0, x ∈ Rd ) satisfies the “S”
property. A proof analogous to that of Lemma 4.5 with un−1 replaced by v shows that the process
(un (t, x), v(t, x), t > 0, x ∈ Rd ) also satisfies the “S” property. Then α(un (t, ·)) − α(v(t, ·)) and
β(un (t, ·)) − β(v(t, ·)) have spatially homogeneous covariances. This ensures that the stochastic
integrals below are well defined. We have
E[(un (t, x) − v(t, x))2 ] 6 2A(t, x) + 2B(t, x),
648
where
An (t, x) = E
"µZ Z
t
Rd
0
and
Bn (t, x) = E
"µZ Z
t
0
¶2 #
Γ(t − s, x − y)(α(un (t, x)) − α(v(t, x)))M (ds, dy)
Γ(t − s, x − y)(β(un (t, x)) − β(v(t, x)))ds dy
Rd
¶2 #
.
Clearly,
An (t, x)
Z t
Z
ds sup E[(un−1 (t, x) − v(t, x))2 ] sup
6C
0
x∈Rd
η∈Rd
Setting
µ(dξ) |FΓ(t − s, ·)(ξ + η)|2 .
(4.10)
Rd
˜ n (t) = sup E[(un (t, x) − v(t, x))2 ]
M
x∈Rd
and using the notations in the proof of Theorem 4.2 we obtain, by (4.10),
Z
An (t, x) 6
t
˜ n−1 (s)J1 (t − s)ds.
M
0
Moreover,
Bn (t, x) 6 C
Z
t
0
ds sup E[(un−1 (t, x) − v(t, x))2 ] sup |FΓ(t − s, ·)(η)|2 ,
x∈Rd
(4.11)
η∈Rd
so
Z
Bn (t, x) 6
t
˜ n−1 (s)J2 (t − s)ds.
M
0
Hence,
˜ n (t) 6
M
Z
t
˜ n−1 (s)J(t − s)ds.
M
0
By [1, Lemma 15], this implies that
˜ n (t) 6
M
Ã
where (an )n∈N is a sequence such that
Finally, we conclude that
2
sup sup E[v(s, x) ]
06s6t x∈Rd
P∞
n=0 an
!
an ,
˜ n (t) → 0 as n → ∞.
< ∞. This shows that M
E[(u(t, x) − v(t, x))2 ] 6 2E[(u(t, x) − un (t, x))2 ] + 2E[(un (t, x) − v(t, x))2 ] → 0,
as n → ∞. This establishes the theorem.
(4.12)
¥
649
5
The non-linear wave equation
As an application of Theorem 4.2, we check the different assumptions in the case of the nonlinear stochastic wave equation in dimensions greater than 3. The case of dimensions 1, 2 and
3 has been treated in [1]. We are interested in the equation
∂2u
− ∆u = α(u)F˙ + β(u),
∂t2
(5.1)
with vanishing initial conditions, where t > 0, x ∈ Rd with d > 3 and F˙ is the noise presented
in Section 2. In the case of the wave operator, the fundamental solution (see [10, Chap.5]) is
d
2π 2
Γ(t) = d 1{t>0}
γ( 2 )
d
2π 2
Γ(t) = d 1{t>0}
γ( 2 )
µ
µ
1∂
t ∂t
1∂
t ∂t
¶ d−2
2
¶ d−3
2
σtd
,
t
if d is odd,
−1
(t2 − |x|2 )+ 2 ,
(5.2)
if d is even,
(5.3)
where σtd is the Hausdorff surface measure on the d-dimensional sphere of radius t and γ is
Euler’s gamma function. The action of Γ(t) on a test function is explained in (5.6) and (5.7)
below. It is also well-known (see [23, §7]) that
FΓ(t)(ξ) =
sin(2πt|ξ|)
,
2π|ξ|
in all dimensions. Hence, there exist constants C1 and C2 , depending on T , such that for all
s ∈ [0, T ] and ξ ∈ Rd ,
sin2 (2πs|ξ|)
C2
C1
6
6
.
(5.4)
2
2
2
1 + |ξ|
4π |ξ|
1 + |ξ|2
Theorem 5.1. Let d > 1, and suppose that
Z
µ(dξ)
sup
< ∞.
2
η∈Rd Rd 1 + |ξ + η|
(5.5)
Then equation (5.1), with α and β Lipschitz functions, admits a random-field solution
(u(t, x), 0 6 t 6 T, x ∈ Rd ). In addition, the uniqueness statement of Theorem 4.8 holds.
Proof. We are going to check that the assumptions of Theorem 4.2 are satisfied. The estimates
in (5.4) show that Γ satisfies (4.4) since (5.5) holds. This condition can be shown to be equivalent
R µ(dξ)
to the condition (40) of Dalang [1], namely Rd 1+|ξ|
2 < ∞ since f > 0 (see [4, Lemma 8] and
[14]). Moreover, taking the supremum over ξ in (5.4) shows that (4.5) is satisfied.
To check
(H1), and in particular, (3.3) and (3.4), fix ϕ ∈ D(Rd ) such that ϕ > 0, supp ϕ ⊂ B(0, 1)
R
and Rd ϕ(x) dx = 1. From formulas (5.2) and (5.3), if d is odd, then
(Γ(t − s) ∗ ϕ)(x) = cd
µ
1 ∂
r ∂r
¶ d−3 "
2
rd−2
650
#¯
¯
¯
(d)
ϕ(x + ry) σ1 (dy) ¯
¯
∂Bd (0,1)
Z
,
r=t−s
(5.6)
(d)
where σ1
is the Hausdorff surface measure on ∂Bd (0, 1), and when d is even,
(Γ(t − s) ∗ ϕ)(x) = cd
µ
1 ∂
r ∂r
¶ d−2 "
2
rd−2
Z
Bd (0,1)
#¯
¯
¯
p
.
ϕ(x + ry) ¯
2
¯
1 − |y|
r=t−s
dy
(5.7)
For 0 6 a 6 b 6 T and a 6 t 6 b, this is a uniformly bounded C ∞ -function of x, with support
contained in B(0, T + 1), and (3.3) and (3.4) clearly hold. Indeed, (Γ(t − s) ∗ ϕ)(x) is always
a sum of products of a positive power of r and an integral of the same form as above but with
respect to the derivatives of ϕ, evaluated at r = t − s. This proves Theorem 5.1.
¥
Remark 5.2. When f (x) = kxk−β , with 0 < β < d, then (5.5) holds if and only if 0 < β < 2.
6
Moments of order p of the solution (p > 2) : the case of affine
multiplicative noise
In the preceding sections, we have seen that the stochastic integral constructed in Section 3
can be used to obtain a random field solution to the non-linear stochastic wave equation in
dimensions greater than 3 (Sections 4 and 5). As for the stochastic integral proposed in [1], this
stochastic integral is square-integrable if the process Z used as integrand is square-integrable.
This property makes it possible to show that the solution u(t, x) of the non-linear stochastic
wave equation is in L2 (Ω) in any dimension.
Theorem 5 in [1] states that Dalang’s stochastic integral is Lp -integrable if the process Z is.
We would like to extend this result to our generalization of the stochastic integral, even though
the approach used in the proof of Theorem 5 in [1] fails in our case. Indeed, that approach is
strongly based on H¨
older’s inequality which can be used when the Schwartz distribution S is
non-negative.
The main interest of a result concerning Lp -integrability of the stochastic integral is to show
that the solution of an s.p.d.e. admits moments of any order and to deduce H¨older-continuity
properties. The first question is whether the solution of the non-linear stochastic wave equation
admits moments of any order, in any dimension ? We are going to prove that this is indeed
the case for a particular form of the no
i
on
Electr
o
u
a
rn l
o
f
P
c
r
ob
abil
ity
Vol. 13 (2008), Paper no. 22, pages 629–670.
Journal URL
http://www.math.washington.edu/~ejpecp/
The non-linear stochastic wave equation in high
dimensions
Daniel Conus and Robert C. Dalang∗
Institut de math´ematiques
Ecole Polytechnique F´ed´erale de Lausanne
Station 8
CH-1015 Lausanne
Switzerland
daniel.conus@epfl.ch, robert.dalang@epfl.ch
Abstract
We propose an extension of Walsh’s classical martingale measure stochastic integral that
makes it possible to integrate a general class of Schwartz distributions, which contains the
fundamental solution of the wave equation, even in dimensions greater than 3. This leads to
a square-integrable random-field solution to the non-linear stochastic wave equation in any
dimension, in the case of a driving noise that is white in time and correlated in space. In the
particular case of an affine multiplicative noise, we obtain estimates on p-th moments of the
solution (p > 1), and we show that the solution is H¨older continuous. The H¨older exponent
that we obtain is optimal .
Key words: Martingale measures, stochastic integration, stochastic wave equation, stochastic partial differential equations, moment formulae, H¨older continuity.
AMS 2000 Subject Classification: Primary 60H15; Secondary: 60H20, 60H05.
Submitted to EJP on February 21, 2007, final version accepted April 1, 2008.
∗
Partially supported by the Swiss National Foundation for Scientific Research.
629
1
Introduction
In this paper, we are interested in random field solutions to the stochastic wave equation
∂2
u(t, x) − ∆u(t, x) = α(u(t, x))F˙ (t, x) + β(u(t, x)),
∂t2
t > 0, x ∈ Rd ,
(1.1)
with vanishing initial conditions. In this equation, d > 1, ∆ denotes the Laplacian on Rd , the
functions α, β : R → R are Lipschitz continuous and F˙ is a spatially homogeneous Gaussian
noise that is white in time. Informally, the covariance functional of F˙ is given by
E[F˙ (t, x)F˙ (s, y)] = δ(t − s)f (x − y),
s, t > 0, x, y ∈ Rd ,
where δ denotes the Dirac delta function and f : Rd → R+ is continuous on Rd \ {0} and even.
We recall that a random field solution to (1.1) is a family of random variables (u(t, x), t ∈
R+ , x ∈ Rd ) such that (t, x) 7→ u(t, x) from R+ × Rd into L2 (Ω) is continuous and solves
an integral form of (1.1): see Section 4. Having a random field solution is interesting if, for
instance, one wants to study the probability density function of the random variable u(t, x)
for each (t, x), as in [12]. A different notion is the notion of function-valued solution, which
is a process t → u(t) with values in a space such as L2 (Ω, L2loc (Rd , dx)) (see for instance [7],
[4]). In some cases, such as [6], a random field solution can be obtained from a function-valued
solution by establishing (H¨
older) continuity properties of (t, x) 7→ u(t, x), but such results are
not available for the stochastic wave equation in dimensions d > 4. In other cases (see [3]), the
two notions are genuinely distinct (since the latter would correspond to (t, x) 7→ u(t, x) from
R+ × Rd into L2 (Ω) is merely measurable), and one type of solution may exist but not the other.
We recall that function-valued solutions to (1.1) have been obtained in all dimensions [14] and
that random field solutions have only been shown to exist when d ∈ {1, 2, 3} (see [1]).
In spatial dimension 1, a solution to the non-linear wave equation driven by space-time white
noise was given in [24], using Walsh’s martingale measure stochastic integral. In dimensions 2
or higher, there is no function-valued solution with space-time white noise as a random input:
some spatial correlation is needed in this case. In spatial dimension 2, a necessary and sufficient
condition on the spatial correlation for existence of a random field solution was given in [2].
Study of the probability law of the solution is carried out in [12].
In spatial dimension d = 3, existence of a random field solution to (1.1) is given in [1]. Since the
fundamental solution in this dimension is not a function, this required an extension of Walsh’s
martingale measure stochastic integral to integrands that are (Schwartz) distributions. This
extension has nice properties when the integrand is a non-negative measure, as is the case for
the fundamental solution of the wave equation when d = 3. The solution constructed in [1] had
moments of all orders but no spatial sample path regularity was established. Absolute continuity
and smoothness of the probability law was studied in [16] and [17] (see also the recent paper
[13]). H¨older continuity of the solution was only recently established in [6], and sharp exponents
were also obtained.
In spatial dimension d > 4, random field solutions were only known to exist in the case of the
linear wave equation (α ≡ 1, β ≡ 0). The methods used in dimension 3 do not apply to higher
dimensions, because for d > 4, the fundamental solution of the wave equation is not a measure,
but a Schwartz distribution that is a derivative of some order of a measure (see Section 5). It
630
was therefore not even clear that the solution to (1.1) should be H¨older continuous, even though
this is known to be the case for the linear equation (see [20]), under natural assumptions on the
covariance function f .
In this paper, we first extend (in Section 3) the construction of the stochastic integral given in
[1], so as to be able to define
Z tZ
S(s, x)Z(s, x)M (ds, dx)
Rd
0
in the case where M (ds, dx) is the martingale measure associated with the Gaussian noise F˙ ,
Z(s, x) is an L2 -valued random field with spatially homogeneous covariance, and S is a Schwartz
distribution, that is not necessarily non-negative (as it was in [1]). Among other technical
conditions, S must satisfy the following condition, that also appears in [14]:
Z
Z t
ds sup
µ(dξ) |FS(s)(ξ + η)|2 < ∞,
0
η∈Rd
Rd
where µ is the spectral measure of F˙ (that is, Fµ = f , where F denotes the Fourier transform).
With this stochastic integral, we can establish (in Section 4) existence of a random field solution
of a wide class of stochastic partial differential equations (s.p.d.e.’s), that contains (1.1) as a
special case, in all spatial dimensions d (see Section 5).
However, for d > 4, we do not know in general if this solution has moments of all orders.
We recall that higher order moments, and, in particular, estimates on high order moments of
increments of a process, are needed for instance to apply Kolmogorov’s continuity theorem and
obtain H¨older continuity of sample paths of the solution.
In Section 6, we consider the special case where α is an affine function and β ≡ 0. This is
analogous to the hyperbolic Anderson problem considered in [5] for d 6 3. In this case, we
show that the solution to (1.1) has moments of all orders, by using a series representation of the
solution in terms of iterated stochastic integrals of the type defined in Section 3.
Finally, in Section 7, we use the results of Section 6 to establish H¨older continuity of the solution
to (1.1) (Propositions 7.1 and 7.2) for α affine and β ≡ 0. In the case where the covariance
function is a Riesz kernel, we obtain the optimal H¨older exponent, which turns out to be the
same as that obtained in [6] for dimension 3.
2
Framework
In this section, we recall the framework in which the stochastic integral is defined. We consider
a Gaussian noise F˙ , white in time and correlated in space. Its covariance function is informally
given by
E[F˙ (t, x)F˙ (s, y)] = δ(t − s)f (x − y),
s, t > 0, x, y ∈ Rd ,
where δ stands for the Dirac delta function and f : Rd → R+ is continuous on Rd \ {0} and
even. Formally, let D(Rd+1 ) be the space of C ∞ -functions with compact support and let F =
{F (ϕ), ϕ ∈ D(Rd+1 )} be an L2 (Ω, F, P)-valued mean zero Gaussian process with covariance
functional
Z
Z ∞ Z
dy ϕ(t, x)f (x − y)ψ(t, y).
dx
dt
E[F (ϕ)F (ψ)] =
0
Rd
Rd
631
Since f is a covariance, there exists a non-negative tempered measure µ whose Fourier transform
is f . That is, for all φ ∈ S(Rd ), the Schwartz space of C ∞ -functions with rapid decrease, we
have
Z
Z
Fφ(ξ)µ(dξ).
f (x)φ(x)dx =
Rd
Rd
As f is the Fourier transform of a tempered measure, it satisfies an integrability condition of
the form
Z
f (x)
dx < ∞,
(2.1)
1
+ |x|p
d
R
for some p < ∞ (see [21, Theorem XIII, p.251]).
Following [2], we extend this process to a worthy martingale measure M = (Mt (B), t > 0, B ∈
Bb (Rd )), where Bb (Rd ) denotes the bounded Borel subsets of R, in such a way that for all
ϕ ∈ S(Rd+1 ),
Z ∞Z
ϕ(t, x)M (dt, dx),
F (ϕ) =
Rd
0
where the stochastic integral is Walsh’s stochastic integral with respect to the martingale measure
M (see [24]). The covariation and dominating measure Q and K of M are given by
Q([0, t] × A × B) = K([0, t] × A × B)
= hM (A), M (B)it = t
Z
Rd
dx
Z
dy 1A (x)f (x − y)1B (y).
Rd
We consider the filtration Ft given by Ft = Ft0 ∨ N , where
Ft0 = σ(Ms (B), s 6 t, B ∈ Bb (Rd ))
and N is the σ-field generated by the P-null sets.
Fix T > 0. The stochastic integral of predictable functions g : R+ × Rd × Ω → R such that
kgk+ < ∞, where
kgk2+ = E
·Z
T
ds
Z
dx
Rd
0
Z
Rd
¸
dy |g(s, x, ·)| f (x − y) |g(s, y, ·)| ,
is defined by Walsh (see [24]). The set of such functions is denoted by P+ . Dalang [1] then
introduced the norm k · k0 defined by
kgk20 = E
·Z
T
ds
0
Z
Rd
dx
Z
Rd
¸
dy g(s, x, ·)f (x − y)g(s, y, ·) .
(2.2)
Recall that a function g is called elementary if it is of the form
g(s, x, ω) = 1]a,b] (s)1A (x)X(ω),
(2.3)
where 0 6 a < b 6 T , A ∈ Bb (Rd ), and X is a bounded Fa -measurable random variable. Now
let E be the set of simple functions, i.e., the set of all finite linear combinations of elementary
functions. Since the set of predictable functions such that kgk0 < ∞ is not complete, let P0
632
denote the completion of the set of simple predictable functions with respect to k · k0 . Clearly,
P+ ⊂ P0 . Both P0 and P+ can be identified with subspaces of P, where
n
P :=
t 7→ S(t) from [0, T ] × Ω → S ′ (Rd ) predictable, such that FS(t) is a.s.
a function and kSk0 < ∞} ,
where
kSk20
=E
·Z
T
dt
Z
2
µ(dξ) |FS(t)(ξ)|
Rd
0
¸
(2.4)
.
For S(t) ∈ S(Rd ), elementary properties of convolution and Fourier transform show that (2.2)
and (2.4) are equal. When d > 4, the fundamental solution of the wave equation provides an
example of an element of P0 that is not in P+ (see Section 5).
Consider a predictable process (Z(t, x), 0 6 t 6 T, x ∈ Rd ), such that
sup sup E[Z(t, x)2 ] < ∞.
06t6T x∈Rd
Let M Z be the martingale measure defined by
Z tZ
Z(s, y)M (ds, dy),
MtZ (B) =
0
0 6 t 6 T, B ∈ Bb (Rd ),
B
in which we again use Walsh’s stochastic integral [24]. We would like to give a meaning to
the stochastic integral of a large class of S ∈ P with respect to the martingale measure M Z .
Following the same idea as before, we will consider the norms k · k+,Z and k · k0,Z defined by
¸
·Z T Z
Z
2
dy |g(s, x, ·)Z(s, x)f (x − y)Z(s, y)g(s, y, ·)|
dx
kgk+,Z = E
ds
and
kgk20,Z
=E
·Z
T
ds
0
Rd
Rd
0
Z
dx
Rd
Z
¸
dy g(s, x, ·)Z(s, x)f (x − y)Z(s, y)g(s, y, ·) .
Rd
(2.5)
Let P+,Z be the set of predictable functions g such that kgk+,Z < ∞. The space P0,Z is
defined, similarly to P0 , as the completion of the set of simple predictable functions, but taking
completion with respect to k · k0,Z instead of k · k0 .
For g ∈ E, as in (2.3), the stochastic integral g · M Z = ((g · M Z )t , 0 6 t 6 T ) is the squareintegrable martingale
Z tZ
Z
Z
g(s, y, ·)Z(s, y)M (ds, dy).
(g · M Z )t = Mt∧b
(A) − Mt∧a
(A) =
0
Rd
Notice that the map g 7→ g ·M Z , from (E, k·k0,Z ) into the Hilbert space M of continuous square1
integrable (Ft )-martingales X = (Xt , 0 6 t 6 T ) equipped with the norm kXk = E[XT2 ] 2 , is an
isometry. Therefore, this isometry can be extended to an isometry S 7→ S·M Z from (P0,Z , k·k0,Z )
into M. The square-integrable martingale S · M Z = ((S · M Z )t , 0 6 t 6 T ) is the stochastic
integral process of S with respect to M Z . We use the notation
Z tZ
S(s, y)Z(s, y)M (ds, dy)
0
Rd
for (S · M Z )t .
The main issue is to identify elements of P0,Z . We address this question in the next section.
633
3
Stochastic Integration
In this section, we extend Dalang’s result concerning the class of Schwartz distributions for which
the stochastic integral with respect to the martingale measure M Z can be defined, by deriving
a new inequality for this integral. In particular, contrary to [1, Theorem 2], the result presented
here does not require that the Schwartz distribution be non-negative.
In Theorem 3.1 below, we show that the non-negativity assumption can be removed provided
the spectral measure satisfies the condition (3.6) below, which already appears in [14] and [4].
As in [1, Theorem 3], an additional assumption similar to [1, (33), p.12] is needed (hypothesis
(H2) below). This hypothesis can be replaced by an integrability condition (hypothesis (H1)
below).
Suppose Z is a process such that sup06s6T E[Z(s, 0)2 ] < +∞ and with spatially homogeneous
covariance, that is z 7→ E[Z(t, x)Z(t, x + z)] does not depend on x. Following [1, Theorem 3],
set f Z (s, x) = f (x)gs (x), where gs (x) = E[Z(s, 0)Z(s, x)].
For s fixed, the function gs is non-negative definite, since it is a covariance function. Hence,
there exists a non-negative tempered measure νsZ such that gs = FνsZ . Note that νsZ (Rd ) =
gs (0) = E[Z(s, 0)2 ]. Using the convolution property of the Fourier transform, we have
f Z (s, ·) = f · gs = Fµ · FνsZ = F(µ ∗ νsZ ),
where ∗ denotes convolution. Looking back to the definition of k · k0,Z , we obtain, for a deterministic ϕ ∈ P0,Z with ϕ(t, ·) ∈ S(Rd ) for all 0 6 t 6 T (see [1, p.10]),
Z T Z
Z
kϕk20,Z =
ds
dx
dy ϕ(s, x)f (x − y)gs (x − y)ϕ(s, y)
0
Rd
Rd
Z T Z
(µ ∗ νsZ )(dξ) |Fϕ(s, ·)(ξ)|2
ds
=
d
R
0
Z
Z T Z
Z
µ(dξ) |Fϕ(s, ·)(ξ + η)|2 .
(3.1)
νs (dη)
ds
=
Rd
Rd
0
In particular,
kϕk20,Z
6
Z
T
0
6 C
Z
ds νsZ (Rd )
T
ds sup
0
η∈Rd
sup
η∈Rd
Z
Z
µ(dξ) |Fϕ(s, ·)(ξ + η)|2
Rd
µ(dξ) |Fϕ(s, ·)(ξ + η)|2 ,
(3.2)
Rd
where C = sup06s6T E[Z(s, 0)2 ] < ∞ by assumption. Taking (3.1) as the definition of k · k0,Z ,
we can extend this norm to the set P Z , where
n
P Z :=
t 7→ S(t) from [0, T ] → S ′ (Rd ) deterministic, such that FS(t) is
a function and kSk0,Z < ∞} .
The spaces P+,Z and P0,Z will now be considered as subspaces of P Z . Let S ∈ P Z . We will
need the following two hypotheses to state the next theorem. Let B(0, 1) denote the open ball
in Rd that is centered at 0 with radius 1.
634
(H1) For all ϕ ∈ D(Rd ) such that ϕ > 0, supp(ϕ) ⊂ B(0, 1), and
0 6 a 6 b 6 T , we have
Z b
(S(t) ∗ ϕ)(·) dt ∈ S(Rd ),
R
Rd
ϕ(x)dx = 1, and for all
(3.3)
a
and
Z
dx
Rd
Z
T
ds |(S(s) ∗ ϕ)(x)| < ∞.
(3.4)
0
(H2) The function FS(t) is such that
lim
h↓0
Z
T
ds sup
η∈Rd
0
Z
|FS(r)(ξ + η) − FS(s)(ξ + η)|2 = 0.
sup
µ(dξ)
Rd
(3.5)
s 1, take
ψn (x) = nd ψ(nx). Then ψn → δ0 in S ′ (Rd ) as n → ∞. Moreover, Fψn (ξ) = Fψ( nξ ) and
|Fψn (ξ)| 6 1, for all ξ ∈ Rd . Define Sn (t) = (ψn ∗ S)(t). As S(t) is of rapid decrease, we have
Sn (t) ∈ S(Rd ) (see [21], Chap. VII, §5, p.245).
Suppose that Sn ∈ P0,Z for all n. Then
kSn −
Sk20,Z
=
Z
T
ds
Rd
0
=
Z
0
Z
T
ds
Z
Rd
νsZ (dη)
νsZ (dη)
Z
Z
µ(dξ) |F(Sn (s) − S(s))(ξ + η)|2
Rd
µ(dξ) |Fψn (ξ + η) − 1|2 |FS(s)(ξ + η)|2 .
Rd
635
(3.8)
The expression |Fψn (ξ + η) − 1|2 is bounded by 4 and goes to 0 as n → ∞ for every ξ and η. By
(3.6), the Dominated Convergence Theorem shows that kSn − Sk0,Z → 0 as n → ∞. As P0,Z is
complete, if Sn ∈ P0,Z for all n, then S ∈ P0,Z .
To complete the proof, it remains to show that Sn ∈ P0,Z for all n.
First consider assumption (H2). In this case, the proof that Sn ∈ P0,Z is based on the same
approximation as in [1]. For n fixed, we can write Sn (t, x) because Sn (t) ∈ S(Rd ) for all
0 6 t 6 T . The idea is to approximate Sn by a sequence of elements of P+,Z . For all m > 1, set
Sn,m (t, x) =
m −1
2X
Sn (tk+1
(t),
m , x)1[tk ,tk+1
m [
m
(3.9)
k=0
where tkm = kT 2−m . Then Sn,m (t, ·) ∈ S(Rd ). We now show that Sn,m ∈ P+,Z . Being a
deterministic function, Sn,m is predictable. Moreover, using the definition of k · k+,Z and the
fact that |gs (x)| 6 C for all s and x, we have
Z
Z T Z
2
dy |Sn,m (s, x)| f (x − y) |gs (x − y)||Sn,m (s, y)|
dx
ds
kSn,m k+,Z =
=
Rd
0
m −1 Z k+1
2X
tm
Rd
ds
tkm
k=0
m −1 Z k+1
2X
tm
6 C
k=0
tkm
Z
dx
Rd
Rd
ds
Z
Z
Rd
k+1
dy |Sn (tk+1
m , x)| f (x − y) |gs (x − y)| |Sn (tm , y)|
˜ k+1
dz f (z)(|Sn (tk+1
m , ·)| ∗ |Sn (tm , ·)|)(z),
k+1
where S˜n (tk+1
m , x) = Sn (tm , −x). By Leibnitz’ formula (see [22], Ex. 26.4, p.283), the function
k+1
k+1
˜
z 7→ (|Sn (tm , ·)| ∗ |Sn (tm , ·)|)(z) decreases faster than any polynomial in |z|−1 . Therefore, by
(2.1), the preceding expression is finite and kSn,m k+,Z < ∞, and Sn,m ∈ P+,Z ⊂ P0,Z .
The sequence of elements of P+,Z that we have constructed converges in k · k0,Z to Sn . Indeed,
Z
Z T Z
µ(dξ) |F(Sn,m (s, ·) − Sn (s, ·))(ξ + η)|2
νsZ (dη)
ds
kSn,m − Sn k20,Z =
d
d
R
R
0
Z
Z T Z
µ(dξ)
sup
νsZ (dη)
ds
|F(Sn (r, ·) − Sn (s, ·))(ξ + η)|2 ,
6
0
Rd
Rd
s0 is a martingale. Moreover,
2
sup Eµ(dξ)×νsZ (dη)×ds [Xm
] 6 Eµ(dξ)×νsZ (dη)×ds [X 2 ] < ∞.
m
The martingale L2 -convergence theorem then shows that (3.12) goes to 0 as m → ∞ and hence
that Sn ∈ P0,Z .
Now, by the isometry property of the stochastic integral between P0,Z and the set M2 of squareintegrable martingales, (S · M Z )t is well-defined and
E[(S · M Z )2T ] = kSk20,Z =
Z
T
ds
0
Z
Rd
νsZ (dη)
Z
µ(dξ) |FS(s, ·)(ξ + η)|2 .
Rd
The bound in the second part of (3.7) is obtained as in (3.2). The result is proved.
¥
Remark 3.2. As can be seen by inspecting the proof, Theorem 3.1 is still valid if we replace
(H2) by the following assumptions :
• t 7→ FS(t)(ξ) is continuous in t for all ξ ∈ Rd ;
• there exists a function t 7→ k(t) with values in the space Sr′ (Rd ) such that, for all 0 6 t 6 T
and h ∈ [0, ε],
|FS(t + h)(ξ) − FS(t)(ξ)| 6 |Fk(t)(ξ)|,
and
Z
T
ds sup
0
η∈Rd
Z
µ(dξ) |Fk(s)(ξ + η)|2 < +∞.
Rd
Remark 3.3. There are two limitations to our construction of the stochastic integral in Theorem
3.1. The first concerns stationarity of the covariance of Z. Under certain conditions (which, in
the case where S is the fundamental solution of the wave equation, only hold for d 6 3), Nualart
and Quer-Sardanyons [13] have removed this assumption. The second concerns positivity of the
covariance function f . A weaker condition appears in [14], where function-valued solutions are
studied.
Integration with respect to Lebesgue measure
In addition to the stochastic integral defined above, we will have to define the integral of the
product of a Schwartz distribution and a spatially homogeneous process with respect to Lebesgue
measure. More precisely, we have to give a precise definition to the process informally given by
Z t Z
dy S(s, y)Z(s, y),
ds
t 7→
0
Rd
where t 7→ S(t) is a deterministic function with values is the space of Schwartz distributions
with rapid decrease and Z is a stochastic process, both satisfying the assumptions of Theorem
3.1.
638
In addition, suppose first that S ∈ L2 ([0, T ], L1 (Rd )). By H¨older’s inequality, we have
"µZ
¶2 #
Z
T
dx |S(s, x)||Z(s, x)|
E
ds
0
6 CE
6 C
6 C
Z
Z
"Z
Rd
T
ds
dx |S(s, x)||Z(s, x)|
Rd
0
T
ds
0
T
ds
0
µZ
Z
Z
dx |S(s, x)|
Z
¶2 #
dy |S(s, y)| E[|Z(s, x)||Z(s, y)|]
Rd
Rd
dx |S(s, x)|
Z
dy |S(s, y)| < ∞,
(3.13)
Rd
Rd
RT R
by the assumptions on Z. Hence 0 ds Rd dx |S(s, x)||Z(s, x)| < ∞ a.s. and the process
Z t Z
dx S(s, x)Z(s, x),
t > 0,
ds
Rd
0
is a.s. well-defined as a Lebesgue-integral. Moreover,
"µZ
¶2 #
Z
T
0 6 E
ds
dx S(s, x)Z(s, x)
=
Z
T
ds
Z
T
ds
=
0
dx
Z
T
ds
Z
Rd
Z
dy S(s, x)S(s, y) E[Z(s, x)Z(s, y)]
Rd
dx
Rd
0
Z
Z
Rd
0
=
Rd
0
Z
dy S(s, x)S(s, y)gs (x − y)
Rd
νsZ (dη) |FS(s)(η)|2 ,
(3.14)
where νsZ is the measure such that FνsZ = gs . Let us define a norm k · k1,Z on the space P Z by
Z T Z
νsZ (dη) |FS(s)(η)|2 .
(3.15)
ds
kSk21,Z =
Rd
0
This norm is similar to k·k0,Z , but with µ(dξ) = δ0 (dξ). In order to establish the next proposition,
we will need the following assumption.
(H2*) The function FS(s) is such that
Z T
ds sup sup
lim
h↓0
0
|FS(r)(η) − FS(s)(η)|2 = 0.
(3.16)
η∈Rd s0 is not necessarily positive and the argument
above does not apply. We need to know a priori that the processes Z(t, x) = α(un (t, x)) −
α(v(t, x)) and W (t, x) = β(un (t, x)) − β(v(t, x)) have a spatially homogeneous covariance. This
is why we consider the restricted class of processes satisfying property “S”.
As u0 ≡ 0, it is clear that the joint process (u0 (t, x), v(t, x), t > 0, x ∈ Rd ) satisfies the “S”
property. A proof analogous to that of Lemma 4.5 with un−1 replaced by v shows that the process
(un (t, x), v(t, x), t > 0, x ∈ Rd ) also satisfies the “S” property. Then α(un (t, ·)) − α(v(t, ·)) and
β(un (t, ·)) − β(v(t, ·)) have spatially homogeneous covariances. This ensures that the stochastic
integrals below are well defined. We have
E[(un (t, x) − v(t, x))2 ] 6 2A(t, x) + 2B(t, x),
648
where
An (t, x) = E
"µZ Z
t
Rd
0
and
Bn (t, x) = E
"µZ Z
t
0
¶2 #
Γ(t − s, x − y)(α(un (t, x)) − α(v(t, x)))M (ds, dy)
Γ(t − s, x − y)(β(un (t, x)) − β(v(t, x)))ds dy
Rd
¶2 #
.
Clearly,
An (t, x)
Z t
Z
ds sup E[(un−1 (t, x) − v(t, x))2 ] sup
6C
0
x∈Rd
η∈Rd
Setting
µ(dξ) |FΓ(t − s, ·)(ξ + η)|2 .
(4.10)
Rd
˜ n (t) = sup E[(un (t, x) − v(t, x))2 ]
M
x∈Rd
and using the notations in the proof of Theorem 4.2 we obtain, by (4.10),
Z
An (t, x) 6
t
˜ n−1 (s)J1 (t − s)ds.
M
0
Moreover,
Bn (t, x) 6 C
Z
t
0
ds sup E[(un−1 (t, x) − v(t, x))2 ] sup |FΓ(t − s, ·)(η)|2 ,
x∈Rd
(4.11)
η∈Rd
so
Z
Bn (t, x) 6
t
˜ n−1 (s)J2 (t − s)ds.
M
0
Hence,
˜ n (t) 6
M
Z
t
˜ n−1 (s)J(t − s)ds.
M
0
By [1, Lemma 15], this implies that
˜ n (t) 6
M
Ã
where (an )n∈N is a sequence such that
Finally, we conclude that
2
sup sup E[v(s, x) ]
06s6t x∈Rd
P∞
n=0 an
!
an ,
˜ n (t) → 0 as n → ∞.
< ∞. This shows that M
E[(u(t, x) − v(t, x))2 ] 6 2E[(u(t, x) − un (t, x))2 ] + 2E[(un (t, x) − v(t, x))2 ] → 0,
as n → ∞. This establishes the theorem.
(4.12)
¥
649
5
The non-linear wave equation
As an application of Theorem 4.2, we check the different assumptions in the case of the nonlinear stochastic wave equation in dimensions greater than 3. The case of dimensions 1, 2 and
3 has been treated in [1]. We are interested in the equation
∂2u
− ∆u = α(u)F˙ + β(u),
∂t2
(5.1)
with vanishing initial conditions, where t > 0, x ∈ Rd with d > 3 and F˙ is the noise presented
in Section 2. In the case of the wave operator, the fundamental solution (see [10, Chap.5]) is
d
2π 2
Γ(t) = d 1{t>0}
γ( 2 )
d
2π 2
Γ(t) = d 1{t>0}
γ( 2 )
µ
µ
1∂
t ∂t
1∂
t ∂t
¶ d−2
2
¶ d−3
2
σtd
,
t
if d is odd,
−1
(t2 − |x|2 )+ 2 ,
(5.2)
if d is even,
(5.3)
where σtd is the Hausdorff surface measure on the d-dimensional sphere of radius t and γ is
Euler’s gamma function. The action of Γ(t) on a test function is explained in (5.6) and (5.7)
below. It is also well-known (see [23, §7]) that
FΓ(t)(ξ) =
sin(2πt|ξ|)
,
2π|ξ|
in all dimensions. Hence, there exist constants C1 and C2 , depending on T , such that for all
s ∈ [0, T ] and ξ ∈ Rd ,
sin2 (2πs|ξ|)
C2
C1
6
6
.
(5.4)
2
2
2
1 + |ξ|
4π |ξ|
1 + |ξ|2
Theorem 5.1. Let d > 1, and suppose that
Z
µ(dξ)
sup
< ∞.
2
η∈Rd Rd 1 + |ξ + η|
(5.5)
Then equation (5.1), with α and β Lipschitz functions, admits a random-field solution
(u(t, x), 0 6 t 6 T, x ∈ Rd ). In addition, the uniqueness statement of Theorem 4.8 holds.
Proof. We are going to check that the assumptions of Theorem 4.2 are satisfied. The estimates
in (5.4) show that Γ satisfies (4.4) since (5.5) holds. This condition can be shown to be equivalent
R µ(dξ)
to the condition (40) of Dalang [1], namely Rd 1+|ξ|
2 < ∞ since f > 0 (see [4, Lemma 8] and
[14]). Moreover, taking the supremum over ξ in (5.4) shows that (4.5) is satisfied.
To check
(H1), and in particular, (3.3) and (3.4), fix ϕ ∈ D(Rd ) such that ϕ > 0, supp ϕ ⊂ B(0, 1)
R
and Rd ϕ(x) dx = 1. From formulas (5.2) and (5.3), if d is odd, then
(Γ(t − s) ∗ ϕ)(x) = cd
µ
1 ∂
r ∂r
¶ d−3 "
2
rd−2
650
#¯
¯
¯
(d)
ϕ(x + ry) σ1 (dy) ¯
¯
∂Bd (0,1)
Z
,
r=t−s
(5.6)
(d)
where σ1
is the Hausdorff surface measure on ∂Bd (0, 1), and when d is even,
(Γ(t − s) ∗ ϕ)(x) = cd
µ
1 ∂
r ∂r
¶ d−2 "
2
rd−2
Z
Bd (0,1)
#¯
¯
¯
p
.
ϕ(x + ry) ¯
2
¯
1 − |y|
r=t−s
dy
(5.7)
For 0 6 a 6 b 6 T and a 6 t 6 b, this is a uniformly bounded C ∞ -function of x, with support
contained in B(0, T + 1), and (3.3) and (3.4) clearly hold. Indeed, (Γ(t − s) ∗ ϕ)(x) is always
a sum of products of a positive power of r and an integral of the same form as above but with
respect to the derivatives of ϕ, evaluated at r = t − s. This proves Theorem 5.1.
¥
Remark 5.2. When f (x) = kxk−β , with 0 < β < d, then (5.5) holds if and only if 0 < β < 2.
6
Moments of order p of the solution (p > 2) : the case of affine
multiplicative noise
In the preceding sections, we have seen that the stochastic integral constructed in Section 3
can be used to obtain a random field solution to the non-linear stochastic wave equation in
dimensions greater than 3 (Sections 4 and 5). As for the stochastic integral proposed in [1], this
stochastic integral is square-integrable if the process Z used as integrand is square-integrable.
This property makes it possible to show that the solution u(t, x) of the non-linear stochastic
wave equation is in L2 (Ω) in any dimension.
Theorem 5 in [1] states that Dalang’s stochastic integral is Lp -integrable if the process Z is.
We would like to extend this result to our generalization of the stochastic integral, even though
the approach used in the proof of Theorem 5 in [1] fails in our case. Indeed, that approach is
strongly based on H¨
older’s inequality which can be used when the Schwartz distribution S is
non-negative.
The main interest of a result concerning Lp -integrability of the stochastic integral is to show
that the solution of an s.p.d.e. admits moments of any order and to deduce H¨older-continuity
properties. The first question is whether the solution of the non-linear stochastic wave equation
admits moments of any order, in any dimension ? We are going to prove that this is indeed
the case for a particular form of the no